Let $A=\{x\ |\ x\in\mathbb Z_{\ge 0},\ x\ $ with some conditions$\ \}$.
Let $B=\mathbb Z_{\ge 0}-A$.
Define $\ 2A= \{a+b : a \in A,\ b \in A\}$.
Define $\ 2B=\{a+b : a \in B,\ b \in B\}$.
Then the set $\ \{n,\ n+1k ,\ n+2k, \ ...\}\ \subseteq\ 2A\ $or $\ 2B$ for some positive integers $n,k$?
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$\begingroup$ No: $A=\{ 2n: n\ge 0 \}$ $\endgroup$– Christian RemlingJun 23, 2014 at 6:01
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$\begingroup$ @ChristianRemling If $\ A=\{2n:n≥0\}\ $,then $B=\{1,3,5,...\},2B = \mathbb Z_{\ge 2}$ for n=2? $\endgroup$– MikeJun 23, 2014 at 6:08
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$\begingroup$ Certainly not, as the sum of two odd numbers is even. $\endgroup$– Christian RemlingJun 23, 2014 at 6:12
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2$\begingroup$ Define $A$ to contain the interval $[2^{2^n},2^{2^{n+1}}]$ if and only if $n$ is odd and positive. We have recently suggested that you stop asking basic questions here, but you've just asked another one. $\endgroup$– S. Carnahan ♦Jun 23, 2014 at 7:44
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1$\begingroup$ What is a "consecutive set"? $\endgroup$– S. Carnahan ♦Jun 23, 2014 at 8:17
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1 Answer
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No. As suggested by S. Carnahan (with the exact numbers tweaked), $$ A = \bigcup_{k=0}^\infty [3^{2k},3^{2k+1}) \quad\text{and}\quad B = \Bbb N\setminus A = \bigcup_{k=0}^\infty [3^{2k+1},3^{2k+2}). $$ Then $$ 2A \subset \bigcup_{k=0}^\infty [3^{2k},2\cdot3^{2k+1}) \quad\text{and}\quad 2B \subset \bigcup_{k=0}^\infty [3^{2k+1},2\cdot3^{2k+2}), $$ and hence both $2A$ and $2B$ have arbitrarily large gaps; this prohibits either set from containing an infinite arithmetic progression.
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$\begingroup$ Thanks,How about $A=\{x\ |\ x\in\mathbb Z_{\ge 0},\ x=\ $some forms$\ \}$? $\endgroup$– MikeJun 23, 2014 at 8:54
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1$\begingroup$ As I read this, I get that one picks c_j and d_j and then puts into A and B integers to ensure that c_j ends up in 2A and d_j ends up in 2B. However, it is also needed for the conclusion that (something like) c_j does not end up in 2B and d_j does not end up in 2A. How do we get this guarantee? $\endgroup$ Jun 23, 2014 at 16:07
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$\begingroup$ @TheMaskedAvenger: hmm, I think you're exactly right. Thanks for pointing out the error. S. Carnahan's comment on the OP seems more correct. (At least this gave me the opportunity to learn that one can't delete an accepted answer....) $\endgroup$ Jun 23, 2014 at 17:47
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1$\begingroup$ At least you can edit it, give Scott's answer with attribution, and assuage your guilt by making the answer CW. $\endgroup$ Jun 23, 2014 at 17:51
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$\begingroup$ Well, I didn't feel all that guilty :) but I still followed your good suggestion. $\endgroup$ Jun 24, 2014 at 22:14